Sparse Coding and Dictionary Learning for Symmetric Positive Definite Matrices: A Kernel Approach

Mehrtash T. Harandi^1,2, Conrad Sanderson^1,2, Richard Hartley^3,4, and Brian C. Lovell^1,2

¹NICTA, PO Box 6020, St Lucia, QLD 4067, Australia

²University of Queensland, School of ITEE, QLD 4072, Australia

³NICTA, Locked Bag 8001, Canberra, ACT 2601, Australia

⁴Australian National University, Canberra, ACT 0200, Australia

Abstract. Recent advances suggest that a wide range of computer vision problems can be addressed more appropriately by considering non-Euclidean geometry. This paper tackles the problem of sparse coding and dictionary learning in the space of symmetric positive definite matrices, which form a Riemannian manifold. With the aid of the recently introduced Stein kernel (related to a symmetric version of Bregman matrix divergence), we propose to perform sparse coding by embedding Riemannian manifolds into reproducing kernel Hilbert spaces. This leads to a convex and kernel version of the Lasso problem, which can be solved efficiently. We furthermore propose an algorithm for learning a Riemannian dictionary (used for sparse coding), closely tied to the Stein kernel. Experiments on several classification tasks (face recognition, texture classification, person re-identification) show that the proposed sparse coding approach achieves notable improvements in discrimination accuracy, in comparison to state-of-the-art methods such as tensor sparse coding, Riemannian locality preserving projection, and symmetry-driven accumulation of local features.

LNCS 7573, p. 216 ff.

Full article in PDF | BibTeX

lncs@springer.com
© Springer-Verlag Berlin Heidelberg 2012